On some mixtures of the Kies distribution

Year 2024, Volume: 53 Issue: 5, 1453 - 1483, 15.10.2024

Tsvetelin Zaevski , Nikolay Kyurkchiev

https://doi.org/10.15672/hujms.1482377

Cited By: 1

Abstract

The purpose of this paper is to explore some mixtures, discrete and continuous, based on the Kies distribution. Some conditions for convergence are established. We study the probabilistic properties of these mixtures. Special attention is taken to the so-called Hausdorff saturation. Several models are examined in detail -- bimodal, multimodal, and mixtures based on binomial, geometric, exponential, gamma, and beta distributions. We provide some numerical experiments for real-life tasks -- one for the Standard and Poor's 500 financial index and another for unemployment insurance issues. In addition, we check the consistency of the proposed estimator using generated data of different sizes.

Keywords

Probability mixtures, Compound distributions, Kies distribution, Exponential distribution, Weibull distribution, Hausdorff saturation

References

• [1] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, US Government printing office, 1968
• [2] T.A. Abushal, T.N. Sindhu, S.A. Lone, M.K.H. Hassan and A. Shafiq, Mixture of Shanker Distributions: Estimation, Simulation and Application, Axioms 12 (3), 231, 2023
• [3] A.Z. Afify, A.M. Gemeay, N.M. Alfaer, G.M. Cordeiro and E.H. Hafez, Power- Modified Kies-Exponential Distribution: Properties, Classical and Bayesian Inference with an Application to Engineering Data, Entropy 24 (7), 883, 2022
• [4] Z. Ahmad, E. Mahmoudi, R. Roozegar, M. Alizadeh and A.Z. Afify, A new exponential-X family: modeling extreme value data in the finance sector, Math. Probl. Eng. 2021 (1), 1-14, 2021
• [5] A.A. Al-Babtain, M.K. Shakhatreh, M. Nassar and A.Z. Afify, A new modified Kies family: Properties, estimation under complete and type-II censored samples, and engineering applications, Mathematics 8 (8), 1345, 2020
• [6] M.M. Al Sobhi, The modified Kies–Fréchet distribution: properties, inference and application, AIMS Math. 6, 4691-4714, 2021
• [7] A. Alsubie, Properties and applications of the modified Kies-Lomax distribution with estimation methods, J. Math. 2021 (1), 1-18, 2021
• [8] M.H. Berger and D. Jeulin, Statistical analysis of the failure stresses of ceramic fibres: dependence of the Weibull parameters on the gauge length, diameter variation and fluctuation of defect density, J. Mater. Sci. 38 (13), 2913-2923, 2003
• [9] F. Blasques, J. van Brummelen, P. Gorgi and S.J. Koopman, Maximum likelihood estimation for non-stationary location models with mixture of Normal distributions, J. Econom. 238 (1), 105575, 2024
• [10] G. D’Amico, R. De Blasis and F. Petroni, The mixture transition distribution approach to networks: evidence from stock markets, Phys. A: Stat. Mech. Appl. 632, 129335, 2023
• [11] J. D’Errico, fminsearchbnd.m; fminsearchcon.m, https://www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbndfminsearchcon. MATLAB Central File Exchange, 2024
• [12] F.S. dos Santos, K.K.F. do Nascimento, J. da Silva Jale, S.F.A. Xavier and T.A.E. Ferreira, Brazilian wind energy generation potential using mixtures of Weibull distributions, Renew. Sust. Energ. Rev. 189, 113990, 2024
• [13] W. Emam and Y. Tashkandy, The Weibull claim model: Bivariate extension, Bayesian, and Maximum Likelihood estimations, Math. Probl. Eng. 2022 (1), 8729529
• [14] I.S. Gradshteyn and I.M. Ryzhik, Table of ntegrals, Series, and Products, Academic press, 2014
• [15] M. Hashempour, A weighted Topp-Leone G family of distributions: properties, applications for modelling reliability data and different method of estimation, Hacet. J. Math. Stat. 51 (5), 1420-1441, 2022
• [16] W. He, Z. Ahmad, A.Z. Afify and H. Goual, The arcsine exponentiated-X family: validation and insurance application, Complexity 2020 (1), 1-18, 2020
• [17] Z. Hu, P. Li, D. Follmann and J. Qin, A mixture distribution approach for assessing genetic impact from twin study, Stat. Med. 41 (14), 2513-2522, 2022
• [18] M. Khalid, M. Aslam and T.N. Sindhu, Bayesian analysis of 3-components Kumaraswamy mixture model: Quadrature method vs. Importance sampling, Alex. Eng. J. 59 (4), 2753-2763, 2020
• [19] J.A. Kies, The strength of glass performance, Naval Research Lab Report 5093, 1958
• [20] S. Li, Y. Liu, Y. Sun and Y. Cai, Deep learning-based channel estimation using Gaussian mixture distribution and expectation maximum algorithm, Phys. Commun. 58, 102018, 2023
• [21] Y. Liu, D. Xie, D.A. Edwards and S. Yu, Mixture copulas with discrete margins and their application to imbalanced data, J. Korean Stat. Soc. 52 (4), 878-900, 2023
• [22] S.A. Lone, S. Anwar, T.N. Sindhu and F. Jarad, Some estimation methods for mixture of extreme value distributions with simulation and application in medicine, Results Phys. 37, 105496, 2022
• [23] S.A. Lone, T.N. Sindhu, S. Anwar, M.K.H. Hassan, S.A. Alsahli and T.A. Abushal, On construction and estimation of mixture of Log-Bilal distributions, Axioms 12 (3), 309, 2023
• [24] R. Maiboroda, V. Miroshnychenko and O. Sugakova, Jackknife for nonlinear estimating equations, Mod. Stoch.: Theory Appl. 9 (4), 377-399, 2022
• [25] S. Matsushita, K. Hagiwara, T. Shiota, H. Shimada, K. Kuramoto and Y. Toyokura, Lifetime data analysis of disease and aging by the Weibull probability distribution, J. Clin. Epidemiol. 45 (10), 1165-1175, 1992
• [26] M. Naderi and M.J. Nooghabi, Clustering asymmetrical data with outliers: Parsimonious mixtures of contaminated mean-mixture of normal distributions, J. Comput. Appl. Math. 437, 115433, 2024
• [27] D.E.Y. Sanku, M. Nassarn and D. Kumar, Moments and estimation of reduced Kies distribution based on progressive type-II right censored order statistics, Hacet. J. Math. Stat. 48 (1), 332-350, 2019
• [28] C. Satheesh Kumar and S.H.S. Dharmaja, On some properties of Kies distribution, Metron 72 (1), 97-122, 2014
• [29] C. Satheesh Kumar and S.H.S. Dharmaja, The exponentiated reduced Kies distribution: properties and applications, Commun. Stat. - Theory Methods 46 (17), 8778- 8790, 2017
• [30] C. Satheesh Kumar and S.H.S. Dharmaja, On modified Kies distribution and its applications, J. Stat. Res. 51 (1), 41-60, 2017
• [31] J.V. Seguro and T.W. Lambert, Modern estimation of the parameters of the Weibull wind speed distribution for wind energy analysis, J. Wind Eng. Ind. Aerodyn. 85 (1), 75-84, 2000
• [32] B. Sendov, Hausdorff Approximations, Springer Science & Business Media 50, 1990
• [33] A. Shafiq, A.B. Çolak, C. Swarup, T.N. Sindhu and S.A. Lone, Reliability analysis based on mixture of Lindley distributions with artificial neural network, Adv. Theory Simul. 5 (8), 2200100, 2022
• [34] A. Shafiq, S.A. Lone, T.N. Sindhu, Y. El Khatib, Q.M. Al-Mdallal and T. Muhammad, A new modified Kies Fréchet distribution: applications of mortality rate of Covid-19, Results Phys. 28, 104638, 2021
• [35] A. Shafiq, T.N. Sindhu, S.A. Lone, M.K.H. Hassan and K. Nonlaopon, Mixture of Akash distributions: estimation, simulation and application, Axioms 11 (10), 516, 2022
• [36] A. Shafiq, A.B. Çolak, S.A. Lone, T.N. Sindhu and T. Muhammad, Reliability modeling and analysis of mixture of Exponential distributions using artificial neural network, Math. Meth. Appl. Sci. 47 (5), 3308-3328, 2024
• [37] F.V.J. Silveira, F. Gomes-Silva, C.R. de Brito, J.S. Jale, F.R.S de Gusmão, S.F.A. Xavier-Junior and J.S. Rocha, Modelling wind speed with a Univariate probability distribution depending on two baseline functions, Hacet. J. Math. Stat. 52 (3), 808- 827, 2023
• [38] T.N. Sindhu, Z. Hussain and M. Aslam, On the Bayesian analysis of censored mixture of two Topp-Leone distribution, Sri Lankan J. Appl. Stat. 19 (1), 13-30, 2019
• [39] T.N. Sindhu, Z. Hussain, N. Alotaibi and T. Muhammad, Estimation method of mixture distribution and modeling of COVID-19 pandemic, AIMS Math. 7 (6), 9926- 9956, 2022
• [40] A. Soulimani, M. Benjillali, H. Chergui and D.B. da Costa, Multihop Weibull-fading communications: performance analysis framework and applications, J. Frankl. Inst.- Eng. Appl. Math. 358 (15), 8012-8044, 2021
• [41] M.T. Vasileva, On Topp-Leone-G power series: saturation in the Hausdorff sense and applications, Mathematics 11 (22), 4620, 2023
• [42] M. Vasileva and N. Kyurkchiev, Insuarance Mathematics, Plovdiv University Press (in Bulgarian), 2023
• [43] H. Wang, Tolerance limits for mixture-of-normal distributions with application to COVID-19 data, WIREs Comput. Stat. 15 (6), e1611, 2023
• [44] Y. Wang, Z. Meng, Z. Zhang, M. Xia, L. Xia and W. Li, A regularization algorithm of dynamic light scattering for estimating the particle size distribution of dual-substance mixture in water, Particuology 89, 246-257, 2024
• [45] W. Weibull, A statistical distribution function of wide applicability, J. Appl. Mech. 18 (3), 293-297, 1951
• [46] D.S. Wilks, Rainfall intensity, the Weibull distribution, and estimation of daily surface runoff, J. Appl. Meteorol. Climatol. 28 (1), 52-58, 1989
• [47] A. Yan, J. Guo and D. Wang, Robust stochastic configuration networks for industrial data modelling with Students-t mixture distribution, Inf. Sci. 607, 493-505, 2022
• [48] J. Yazhou, W. Molin and J. Zhixin, Probability distribution of machining center failures, Reliab. Eng. Syst. Saf. 50 (1), 121-125, 1995
• [49] Y.I. Yeleyko and O.A. Yarova, Mixture of distributions based on the Markov chain, Cybern. Syst. Anal. 58 (5), 754-757, 2022
• [50] T.S. Zaevski and N. Kyurkchiev, Some notes on the four-parameters Kies distribution, C. R. Acad. Bulg. Sci. 75 (10), 1403-1409, 2022
• [51] T. Zaevski and N. Kyurkchiev, On some composite Kies families: distributional properties and saturation in Hausdorff sense, Mod. Stoch.: Theory Appl. 10 (3), 287-312, 2023
• [52] T. Zaevski and N. Kyurkchiev, On min- and max-Kies families: distributional properties and saturation in Hausdorff sense, Mod. Stoch.: Theory Appl. 11 (3), 265-288, 2024
• [53] T. Zaevski and N. Kyurkchiev, On the Hausdorff saturation of some trigonometric- Kies families, Palest. J. Math. 13 (2), 249-262, 2024
• [54] Y. Zhang, Y. Dong and R. Feng, Bayes-informed mixture distribution for the EVD estimation and dynamic reliability analysis, Mech. Syst. Signal Proc. 197, 110352, 2023
• [55] Y. Zhenwu, Z. Ahmad, Z. Almaspoor and S.K. Khosa, On the genesis of the Marshall- Olkin family of distributions via the T-X family approach: statistical modeling, CMCComput. Mat. Contin. 67 (1), 753-760, 2021